Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras

2023/10/09 by Ilijas Farah, Farah, Ilijas, David Jekel +3
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA) #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.2310.06197

openalex publication_date 2023/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra N is never model complete if its direct integral decomposition contains II₁ factors M such that M₂(M) embeds into an ultrapower of M. The proof in the case of II₁ factors uses an explicit construction based on random matrices and quantum expanders.

Related

• The theory of tracial von Neumann algebras does not have a model companion
• Quantifier elimination in II₁ factors
• A local quantization principle for inclusions of tracial von Neumann algebras
• A survey on the model theory of tracial von Neumann algebras
• Correspondences, Ultraproducts and Model Theory